Boltzmann equation

Ludwig Eduard Boltzmann is responsible for a lot of equations and in various fields, different equations are termed the Boltzmann equation (see below). In the field of stellar structure, the term (or Boltzmann relation, which is also ambiguous) is used for an equation giving the ratio of counts of atoms at different possible atomic excitation levels. One form:

Ni+1   gi+1
———— = ———— e-(Ei+1-Ei)/kT
 Ni     gi

• Nⁱ⁺¹, Nⁱ - number density of two states of excitation i and i+1.
• gⁱ⁺¹, gⁱ - quantum-mechanical degeneracy weights.
• k - Boltzmann constant.
• T - temperature.

This equation is related to the Boltzmann distribution (frequency of particles at various states, not the same as the Maxwell-Boltzmann distribution. which is the distribution of (say) the kinetic energy of particles in (say) a gas) incorporating a Boltzmann factor (e^{-(E₁-E₂)/kT}) giving the relative probability of two energy levels, and relating the ratio of the two terms, N_x/g_x (the number of atoms in specific quantum state) to it. This Boltzmann equation is analogous to the Saha equation which gives similar ratios for states of ionization.

Another Boltzmann equation used in astrophysics is the Boltzmann transport equation (BTE, often shortened to Boltzmann equation), which is more general than the Maxwell-Boltzmann distribution, handling gases not necessarily in thermodynamic equilibrium. (It seems plausible to me that the above relation is derived from this, a possible reason for the "name clash".)